A study for which only half turned up (3)
I believe the answer is:
den
'a study' is the definition.
(term for a study or office)
'only half turned up' is the wordplay.
'only half' suggests halving the letters.
'up' is a reversal indicator (in down clue: letters go upwards).
'turned' halved is 'ned'.
'ned' written backwards gives 'DEN'.
'for which' acts as a link.
(Other definitions for den that I've seen before include "Miserable place" , "Animal lair or gent's study" , "Animal habitation" , "Animal's home or gent's study" , "Hideaway" .)